Skills in Context

I’ve been bothered for a long time by a common practice in math textbooks and classes. Because math is a logical system, it makes sense to write it out in logical order: start with the basics (definitions) and build up logically to the more complex results and applications. This makes sense for a rigorous math reference book. Even most high school textbooks reflect this order, and unfortunately, so do most teachers.

But is this really the order that makes sense for a first time learner? Why should they pay attention to some technical definitions when they have no idea why they should be able to work with those definitions? How exciting is it for most people to learn math this way?

I think it’s the wrong way to teach math. I am always looking for ways to create a “need to know” in my students by starting with the “interesting” problems (even if they are currently completely intractable to the students) and using that to motivate learning new definitions and skills.

For example: most of my precalculus students have seen logarithms, but so briefly that they barely remember the word logarithm (and that they don’t like it). So when we approach logarithms in precalculus, I like to start by working with exponential modeling problems. I ask students something like, “Okay, so you can tell me the population of bacteria at any given time. But can you tell me at what time the population will reach 8000?”

Solving this analytically requires logarithms, but they don’t remember that. So I let them try any method they can… guessing, using the graph or table of a calculator, or even trying some illegal analytic methods (taking the “nth” root of both sides). But then I try to instill that it’s bothersome that we can’t solve this by hand. I find problems that circumvent calculator methods or show them that guess and check is too inaccurate to be realistic. That motivates our need for logarithms, because logarithms are the inverse operation of exponentials and allow us to solve these equations.

Contrast this with the four or so precalculus textbooks that I have access to: they all introduce the definitions of exponential and logarithmic functions together and immediately start getting students to rewrite exponents as logarithms and vice versa (simple definitions). They don’t move on to applications or using logs/exponents together as inverse operations until a few sections later! It usually takes until the end of the unit to really “see” how logarithms and exponentials fit in together, and by then, many students are likely already lost and cemented in their lifelong hatred of logarithms.

The contexts don’t always need to be “real-world”, though that can certainly work. Sometimes the context is purely mathematical and helps students see why number theory is so much fun. For instance, my geometry students this year were incredibly intrigued by this question: Given an integer n, you have to create triangles with integer sides whose perimeter is n. For n = 1-10, how many different triangles can be created for each n?

I was honestly surprised by how involved the students got, and multiple students remarked how fun it was to complete a table and look for patterns. It was a great motivation to establish an often boring Triangle Inequality Theorem and I think the intuitive meaning stuck with my students much more than in the past.

I know as a first year teacher I often felt I had no choice but to follow the book, which was incredibly hard with the precalculus book we use because it is written this way. But the longer I teach, the more I am convinced that motivation and interest are more important than almost anything else in math. We need to find ways to introduce topics in context so that students have a reason to learn (other than grades, of course).

Not Doing Their Homework

So, I’ve been seriously questioning the use of homework this year, especially after reading a large amount of Alfie Kohn over the summer. Last year, I had a large group of students who would not do the homework, despite parent interventions, tutoring, advisor interventions, and very lenient deadlines. Most of them did not do well on tests, either, and I found it easy enough to keep pointing to homework as a panacea… “If he would only do his homework, he would understand!”

This year, I have a smaller group that doesn’t do their homework, but they’re different. They’re nerds. They like math, programming, and science. They ask great questions in class, dive into problems readily, come up with really amazing intuitive solutions, explain things to their peers, and strike me as the type of students who really could excel in math or science. But they don’t usually get the homework done… and honestly, I can’t say that they need to. They don’t ace every test, but they care about math and seem to understand the material a lot better than their grades would portray because they’re being punished by missing homework.

There’s something wrong with this. If you observe my class, you would likely pick these students out to be some of the top students, but they’re not (in terms of grades). I think my need for “fairness” and “objectivity” has kept me from seeing before how that homework grade is really affecting many students in the wrong way. On the other hand, I have a small group of students who struggle in class but always do their homework, and their grades are overinflated as a result.

Sadly, my grades don’t seem to be reflecting what students actually know. Giving grades in general is something I also struggle with, but as long as we have to do it, I want the grades to mean something.

Some other great posts about homework that I’ve read recently make me think I’m not completely insane for wanting to give it up:

Tom Whitby: The Homework Option Plan

Dan Meyer: Why I Don’t Assign Homework

Shawn Cornally: Stop Grading Homework, Please

Winning Streaks, Losing Streaks

Another assessment article I dug up recently was “Assessment Through the Student’s Eyes” by Rick Stiggins.

“The goal of assessment for learning is not to eliminate failure, but rather to keep failure from becoming chronic and thus inevitable in the mind of the learner. Duke University basketball coach Mike Krzyzewski has pointed out that the key to winning is to avoid losing twice in a row (Kanter, 2004, p. 251). He meant that if you lose once and fix it, you can remain confident. Losing twice, though, can raise questions, crack that confidence, and make recovery more difficult. So when learners suffer a failure, we must get them back to success as quickly as possible to restore their confidence in their capabilities. This is the emotional dynamic of assessment for learning.”

I love the sports analogy. In something like sports, we so easily make the connection between the emotional state of a team and its successes; any devoted fan will concede a loss here and there, but the more losses happen, the more chronic they usually become.

I feel like teachers don’t really consider the emotional impact of handing back bad grades, or the impact of handing back bad grade after bad grade. Or likewise, they don’t consider the kid has who always scores well and just keeps scoring well; they are both at an emotional advantage and more likely to continue succeeding.

I was a high school student only a mere 6 years ago, so I kind of remember being in that position. I was usually the high achiever though, so it’s hard to remember what it feels like to be a “losing streak”. However, I did always feel that way with regard to math. I was in advanced classes, but math was the hardest subject for me. I “knew” I was never going to do as well as some of my friends in my math classes. I would actually tell people I was “bad at math” even as a junior in Calculus class. Looking back, it’s hard to believe I said that; in college, I actually studied math because I found it challenging and I’d finally found the self-confidence to see that as a good thing.

Maybe it’s the same for a lot of my students. They just keep “losing”, whether losing for them means a D grade or a B grade. The challenge is that I still want assessment to be fair; I don’t want to hand out high grades to everyone just to make them feel good. It would be an interesting experiment, but I think that the students need to feel like they really accomplished the “win”, not just have it given to them. I’ve seen it happen with a few students already this semester… like one who struggled last semester. She did really well on our first test of the semester, and ever since then she keeps proudly telling me how she got all the homework right (rather than her usual “Wait, I’m confused”)… and indeed, her work looks astronomically better.

So what does Stiggens suggest to fairly, and honestly, help more students get on a “winning streak”?

“Assessment for learning begins when teachers share achievement targets with students, presenting those expectations in student-friendly language accompanied by examples of exemplary student work. Then, frequent self-assessments provide students (and teachers) with continual access to descriptive feedback in amounts they can manage effectively without being overwhelmed. Thus, students can chart their trajectory toward the transparent achievement targets their teachers have established. The students’ role is to strive to understand what success looks like, to use feedback from each assessment to discover where they are now in relation to where they want to be, and to determine how to do better the next time. As students become increasingly proficient, they learn to generate their own descriptive feedback and set goals for what comes next on their journey.”

Two specific situations he discusses are:
1. Set Students Up for Success (sharing expectations)
2. Help Students Turn Failure into Success (i.e. test corrections)

I’ve been good at #2. I have already done various types of test corrections and/or relearning type activities. I’m not sure it’s always helped everyone take ownership and learn things better, but I think it has helped some really think about their performance and improve. However, #1 is where I feel I’ve been less strong. It’s not that I want to keep things a secret, but I find that I have given assignments where I expected something and clearly the students didn’t understand that. For example, even with timed testing at my school I ran into a misunderstanding. It’s a small private school and apparently it’s the norm (especially in middle school) to just let students have extra time on tests whenever they want. As for whether timed testing is even the right way to approach things… that’s a whole different discussion. However, given my background, I didn’t think I had to explain ahead of time that a timed test meant you don’t get extra time. I got major backlash from students grades 9-12 who weren’t used to that expectation and even got talked to by the administration. It’s not that they didn’t support me, but students complained. That’s an example of where I could have been clearer with expectations, and I know in the future I will be.

Assessment Group

In preparation for an “Assessment Group” meeting after work today, I’ve been reading about formative assessment.

“What we need is a shift from quality control in learning to quality assurance. Traditional approaches to instruction and assessment involve teaching some given material, and then, at the end of teaching, working out who has and hasn’t learned it–akin to a quality control approach in manufacturing. In contrast, assessment for learning involves adjusting teaching while the learning is still taking place–a quality assurance approach. Quality assurance also involves a shift of attention from teaching to learning. The emphasis is on what the students are getting out of the process rather than on what teachers are putting into it, reminiscent of the old joke that schools are places where children go to watch teachers work.” (from “Classroom Assessment: Minute by Minute, Day by Day” by Leahy, Lyon, Thompson, Wiliam)

I chuckle at the idea of students going to watch teachers work – I do sometimes feel like they put in so little effort and here I am stressing myself out, working at home, having nightmares about school…

I really do agree with the ideas of formative assessment. I guess I just keep finding myself conflicted in how to do it.

For example, I made a quiz over the graphs of exponential functions for my precalculus classes. My whole goal with the quiz was to make questions that could get at the shapes of graphs relative to other graphs but so that the kids couldn’t just plug it into their calculators. It’s like when I get into “test and quiz making” mode, I come up with these tricky nice questions, but I almost wish they weren’t being graded on them. I honestly call it test or quiz mostly so they will take it seriously!

I think they can learn from the questions, but how do I administer the quiz and have them learn from it? I would rather just not grade the quiz at all and use it for discussion, but I feel like they expect a grade and who am I to completely change in the middle of the year?

Never Passing Through the Student’s Mind

“U.S. Senator Jeff Bingaman recalled what his father, a chemistry professor, used to say about traditional lectures: the notes of the teacher go straight to the notes of the student without ever passing through the student’s mind.”

I stumbled on this in an article about a local charter school, and it relates directly to a situation I had yesterday. There’s a 10th grade girl in my geometry class who struggles in math. She told me this at the beginning of the year, and I tried to inspire her that this year could be different. She’s a really interesting girl who perks up at any mention of fractals or Fibonacci and turned in a really eloquent piece of writing for my first foray into making the students write in math class.

So she ends up pulling up to a B- last semester and was absolutely ecstatic. Then yesterday was the first test for this semester. A couple days ago I was having them work on a review of similar triangles while I went around and talked to each of them, and I started to become a little worried about her. Similar triangles usually involves dealing with fractions, and it became clear to me that her fraction skills were shaky at best.

So anyway, yesterday right before the test, she comes into my office and has this big confession to make. She says that she likes my class, but for some reason this whole unit on similar triangles has just not stuck with her. She says that she just feels like no matter how hard she tries to listen, she can’t focus on the lecture. When she can focus, she understands what I’m doing, but then she tunes out again and then when she’s looking at the homework problems, she has no idea what to do. She kept saying that she thought she knew the concepts but “couldn’t express it” the way that I showed, which basically meant she doesn’t solve proportions the way I do and was concerned about having to show her work. When I saw her working in class the other day, she actually knew what the answer was supposed to be but couldn’t explain why to her friend sitting next to her. She asked me “how to do it”, which I interpreted to mean how to set up the algebraic proportion and solve, and that’s where she was really lost.

I encouraged her to take the test and answer the questions “her way” to the best of her ability, not worrying if the “work” looked like what I showed in class. What I saw on her test was actually quite interesting. She was looking at the relationships between the sides in an addition sense – so that if one side increased by 10, she wanted the other side to increase by 10. She obviously missed the entire idea of increasing by a multiplicative factor, and I never managed to figure that out before the test.

So it makes me really wonder what to do now. I don’t feel right giving her a low test grade when I can finally just now see that she has a huge conceptual misunderstanding. Honestly, I’ve been questioning testing in general lately, but this really makes me wonder.

I know that when I give back tests, most students hardly look at it. The ones who scored poorly look embarrassed and are usually the ones that put it away the quickest. I kind of doubt most of them go home and pore over and read the comments and try to figure out how to improve.

I did a test corrections assignment earlier this year with mixed results. I’m still not sure it really shows them what to focus on; they just figure out how to fix that one problem in hopes of getting some points back. I mean after all, it’s hard for someone to look at their own paper and pick out their areas of weakness, but it’s fairly easy for me to do provided they show enough of their thought process in writing.

So I have this idea and I’m a little hesitant to do it because it would take longer.. but I was thinking of writing each student a short summary of their test that mentions the areas of deficiency, and giving them that before they ever see a red pen mark or a grade. Then, those that score below a certain threshold can meet with me to work on that deficiency and possibly earn back some credit.

I don’t know. It’s still playing in the “points game” which I am really starting to hate, but I’m not ready to go completely off the deep end with the anti-testing stuff yet. At least I feel like they’d have more of a chance of getting feedback before they shut off when they see a low grade.

A Mathematician’s Lament

“Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics… Mathematics is the purest of the arts, as well as the most misunderstood.” – Paul Lockhart, A Mathematician’s Lament

I’ve been on a “humanist” math philosophy reading kick lately. I’ve gotten myself partway through two of Reuben Hersh’s books and many of his articles and found many interesting TED talks supporting the view that teaching, regardless of the subject, has changed in many ways and we need to adapt.

Lockhart’s article is one of the better pieces I’ve read on what is apparently a fairly subversive view of mathematics and mathematics education. I sat reading it, completely absorbed, getting more and more fired up with each page. The article addresses ideas which have been circulating in my mind a lot lately; one semester into my teaching career, and I’m already seriously throwing into question 80% of what I thought I knew about math education.

I am a brand new teacher – graduated with a bachelor’s in pure math less than two years ago. I never went to education school, which Lockhart refers to as “a complete crock”. He argues that “teaching is not about information. It’s about having an honest intellectual relationship with your students. It requires no method, no tools, and no training. Just the ability to be real. And if you can’t be real, then you have no right to inflict yourself upon innocent children.”

I can see this pretty clearly in my own experience this year. I teach geometry, which Lockhart calls the “instrument of the Devil” and precalculus, which he refers to as “a senseless bouillabaisse of disconnected topics” (is it bad if that quote makes me hungry?). Just halfway through my first year of teaching each of those courses, I already see frustration in the students with the traditional way of teaching. They shut off completely during geometric “proofs”, which I have now almost completely abandoned after realizing that not only did the students not understand them, but apparently lost the ability to explain simple concepts in their own words because they were so preoccupied with having to write a “geometric two column proof”. Precalculus has lots of review, and many of the students are completely shut off by having the same material they’ve seen before just lectured back to them in about the same depth.

It’s too easy for me to “be a passive conduit of some publisher’s “materials” and to follow the shampoo-bottle instruction “lecture, test, repeat” than to think deeply and thoughtfully about the meaning of one’s subject and how best to convey that meaning directly and honestly to one’s students,” as Lockhart describes it. So easy, and so boring, that in less than one semester, I am already itching to get further away from stand-and-deliver-teaching and the mind-numbing textbooks we use.

“Why don’t we want our children to learn to do mathematics? Is it that we don’t trust them, that we think it’s too hard? We seem to feel that they are capable of making arguments and coming to their own conclusions about Napoleon, why not about triangles? I think it’s simply that we as a culture don’t know what mathematics is. The impression we are given is of something very cold and highly technical, that no one could possibly understand— a self fulfilling prophesy if there ever was one.”

“All this fussing and primping about which “topics” should be taught in what order, or the use of this notation instead of that notation, or which make and model of calculator to use, for god’s sake— it’s like rearranging the deck chairs on the Titanic! Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion— not because it makes no sense to you, but because you gave it sense and you still don’t understand what your creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it.”

Many teachers are afraid to let real math happen. They want to stand and lecture and prevent the kids from trying anything new or ever making any mistakes. It seems they believe math is a set of absolute truths and they think they have been appointed by the gods to stand and deliver these truths to the congregation. Remember the lessons of medieval churches; congregations can’t read the holy word themselves and make sense of it, so we need the holy priests to deliver the word unto them. Looks like we need a real Reformation for math.

I could go on. But I think this is good enough for a first post. I’ll leave you with one last line from Lockhart (who really is a smart cookie, by the way – I laughed, got angry, got inspired, and felt very stimulated by the article).

“It’s perfectly simple. Students are not aliens. They respond to beauty and pattern, and are naturally curious like anyone else. Just talk to them! And more importantly, listen to them!”

If my ramblings have piqued your interest in the article (which is a little long at 25 pages, but truly excellent to read), you can find it online here.